The notion of crossed homomorphisms of Lie algebras was first introduced by Lue [1]} in the study of non-abelian extensions of Lie algebras. A crossed homomorphism is nothing but a differential
operator of weight 1. The authors showed that the category of weak representations
(resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module
category over the monoidal category of representations of Lie algebras using crossed homomorphisms [2]}. Recently, the author considerd crossed homomorphisms between associative algebras [3]}.
